# Properties

 Label 4608.2.a.y.1.2 Level $4608$ Weight $2$ Character 4608.1 Self dual yes Analytic conductor $36.795$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4608 = 2^{9} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4608.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$36.7950652514$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{5})$$ Defining polynomial: $$x^{4} - 6x^{2} + 4$$ x^4 - 6*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$0.874032$$ of defining polynomial Character $$\chi$$ $$=$$ 4608.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.41421 q^{5} +3.16228 q^{7} +O(q^{10})$$ $$q-1.41421 q^{5} +3.16228 q^{7} +4.47214 q^{11} +4.47214 q^{13} +6.32456 q^{17} -2.82843 q^{19} +4.00000 q^{23} -3.00000 q^{25} +4.24264 q^{29} -3.16228 q^{31} -4.47214 q^{35} +4.47214 q^{37} +6.32456 q^{41} -8.48528 q^{43} +12.0000 q^{47} +3.00000 q^{49} -7.07107 q^{53} -6.32456 q^{55} -13.4164 q^{61} -6.32456 q^{65} +8.00000 q^{71} -4.00000 q^{73} +14.1421 q^{77} +3.16228 q^{79} -4.47214 q^{83} -8.94427 q^{85} +14.1421 q^{91} +4.00000 q^{95} +2.00000 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q + 16 q^{23} - 12 q^{25} + 48 q^{47} + 12 q^{49} + 32 q^{71} - 16 q^{73} + 16 q^{95} + 8 q^{97}+O(q^{100})$$ 4 * q + 16 * q^23 - 12 * q^25 + 48 * q^47 + 12 * q^49 + 32 * q^71 - 16 * q^73 + 16 * q^95 + 8 * q^97

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 0 0
$$4$$ 0 0
$$5$$ −1.41421 −0.632456 −0.316228 0.948683i $$-0.602416\pi$$
−0.316228 + 0.948683i $$0.602416\pi$$
$$6$$ 0 0
$$7$$ 3.16228 1.19523 0.597614 0.801784i $$-0.296115\pi$$
0.597614 + 0.801784i $$0.296115\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 4.47214 1.34840 0.674200 0.738549i $$-0.264489\pi$$
0.674200 + 0.738549i $$0.264489\pi$$
$$12$$ 0 0
$$13$$ 4.47214 1.24035 0.620174 0.784465i $$-0.287062\pi$$
0.620174 + 0.784465i $$0.287062\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 6.32456 1.53393 0.766965 0.641689i $$-0.221766\pi$$
0.766965 + 0.641689i $$0.221766\pi$$
$$18$$ 0 0
$$19$$ −2.82843 −0.648886 −0.324443 0.945905i $$-0.605177\pi$$
−0.324443 + 0.945905i $$0.605177\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 4.00000 0.834058 0.417029 0.908893i $$-0.363071\pi$$
0.417029 + 0.908893i $$0.363071\pi$$
$$24$$ 0 0
$$25$$ −3.00000 −0.600000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 4.24264 0.787839 0.393919 0.919145i $$-0.371119\pi$$
0.393919 + 0.919145i $$0.371119\pi$$
$$30$$ 0 0
$$31$$ −3.16228 −0.567962 −0.283981 0.958830i $$-0.591655\pi$$
−0.283981 + 0.958830i $$0.591655\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −4.47214 −0.755929
$$36$$ 0 0
$$37$$ 4.47214 0.735215 0.367607 0.929981i $$-0.380177\pi$$
0.367607 + 0.929981i $$0.380177\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 6.32456 0.987730 0.493865 0.869539i $$-0.335584\pi$$
0.493865 + 0.869539i $$0.335584\pi$$
$$42$$ 0 0
$$43$$ −8.48528 −1.29399 −0.646997 0.762493i $$-0.723975\pi$$
−0.646997 + 0.762493i $$0.723975\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 12.0000 1.75038 0.875190 0.483779i $$-0.160736\pi$$
0.875190 + 0.483779i $$0.160736\pi$$
$$48$$ 0 0
$$49$$ 3.00000 0.428571
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −7.07107 −0.971286 −0.485643 0.874157i $$-0.661414\pi$$
−0.485643 + 0.874157i $$0.661414\pi$$
$$54$$ 0 0
$$55$$ −6.32456 −0.852803
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ −13.4164 −1.71780 −0.858898 0.512148i $$-0.828850\pi$$
−0.858898 + 0.512148i $$0.828850\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −6.32456 −0.784465
$$66$$ 0 0
$$67$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 8.00000 0.949425 0.474713 0.880141i $$-0.342552\pi$$
0.474713 + 0.880141i $$0.342552\pi$$
$$72$$ 0 0
$$73$$ −4.00000 −0.468165 −0.234082 0.972217i $$-0.575209\pi$$
−0.234082 + 0.972217i $$0.575209\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 14.1421 1.61165
$$78$$ 0 0
$$79$$ 3.16228 0.355784 0.177892 0.984050i $$-0.443072\pi$$
0.177892 + 0.984050i $$0.443072\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −4.47214 −0.490881 −0.245440 0.969412i $$-0.578933\pi$$
−0.245440 + 0.969412i $$0.578933\pi$$
$$84$$ 0 0
$$85$$ −8.94427 −0.970143
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 0 0
$$91$$ 14.1421 1.48250
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 4.00000 0.410391
$$96$$ 0 0
$$97$$ 2.00000 0.203069 0.101535 0.994832i $$-0.467625\pi$$
0.101535 + 0.994832i $$0.467625\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −15.5563 −1.54791 −0.773957 0.633238i $$-0.781726\pi$$
−0.773957 + 0.633238i $$0.781726\pi$$
$$102$$ 0 0
$$103$$ −9.48683 −0.934765 −0.467383 0.884055i $$-0.654803\pi$$
−0.467383 + 0.884055i $$0.654803\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −17.8885 −1.72935 −0.864675 0.502331i $$-0.832476\pi$$
−0.864675 + 0.502331i $$0.832476\pi$$
$$108$$ 0 0
$$109$$ 4.47214 0.428353 0.214176 0.976795i $$-0.431293\pi$$
0.214176 + 0.976795i $$0.431293\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −12.6491 −1.18993 −0.594964 0.803752i $$-0.702834\pi$$
−0.594964 + 0.803752i $$0.702834\pi$$
$$114$$ 0 0
$$115$$ −5.65685 −0.527504
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 20.0000 1.83340
$$120$$ 0 0
$$121$$ 9.00000 0.818182
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 11.3137 1.01193
$$126$$ 0 0
$$127$$ 15.8114 1.40303 0.701517 0.712653i $$-0.252506\pi$$
0.701517 + 0.712653i $$0.252506\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 17.8885 1.56293 0.781465 0.623949i $$-0.214473\pi$$
0.781465 + 0.623949i $$0.214473\pi$$
$$132$$ 0 0
$$133$$ −8.94427 −0.775567
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −18.9737 −1.62103 −0.810515 0.585718i $$-0.800813\pi$$
−0.810515 + 0.585718i $$0.800813\pi$$
$$138$$ 0 0
$$139$$ −16.9706 −1.43942 −0.719712 0.694273i $$-0.755726\pi$$
−0.719712 + 0.694273i $$0.755726\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 20.0000 1.67248
$$144$$ 0 0
$$145$$ −6.00000 −0.498273
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −18.3848 −1.50614 −0.753070 0.657941i $$-0.771428\pi$$
−0.753070 + 0.657941i $$0.771428\pi$$
$$150$$ 0 0
$$151$$ 22.1359 1.80140 0.900699 0.434444i $$-0.143055\pi$$
0.900699 + 0.434444i $$0.143055\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 4.47214 0.359211
$$156$$ 0 0
$$157$$ 13.4164 1.07075 0.535373 0.844616i $$-0.320171\pi$$
0.535373 + 0.844616i $$0.320171\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 12.6491 0.996890
$$162$$ 0 0
$$163$$ 19.7990 1.55078 0.775388 0.631485i $$-0.217554\pi$$
0.775388 + 0.631485i $$0.217554\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 12.0000 0.928588 0.464294 0.885681i $$-0.346308\pi$$
0.464294 + 0.885681i $$0.346308\pi$$
$$168$$ 0 0
$$169$$ 7.00000 0.538462
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 7.07107 0.537603 0.268802 0.963196i $$-0.413372\pi$$
0.268802 + 0.963196i $$0.413372\pi$$
$$174$$ 0 0
$$175$$ −9.48683 −0.717137
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 17.8885 1.33705 0.668526 0.743689i $$-0.266925\pi$$
0.668526 + 0.743689i $$0.266925\pi$$
$$180$$ 0 0
$$181$$ 13.4164 0.997234 0.498617 0.866822i $$-0.333841\pi$$
0.498617 + 0.866822i $$0.333841\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −6.32456 −0.464991
$$186$$ 0 0
$$187$$ 28.2843 2.06835
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 20.0000 1.44715 0.723575 0.690246i $$-0.242498\pi$$
0.723575 + 0.690246i $$0.242498\pi$$
$$192$$ 0 0
$$193$$ −16.0000 −1.15171 −0.575853 0.817554i $$-0.695330\pi$$
−0.575853 + 0.817554i $$0.695330\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −7.07107 −0.503793 −0.251896 0.967754i $$-0.581054\pi$$
−0.251896 + 0.967754i $$0.581054\pi$$
$$198$$ 0 0
$$199$$ 3.16228 0.224168 0.112084 0.993699i $$-0.464247\pi$$
0.112084 + 0.993699i $$0.464247\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 13.4164 0.941647
$$204$$ 0 0
$$205$$ −8.94427 −0.624695
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −12.6491 −0.874957
$$210$$ 0 0
$$211$$ −5.65685 −0.389434 −0.194717 0.980859i $$-0.562379\pi$$
−0.194717 + 0.980859i $$0.562379\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 12.0000 0.818393
$$216$$ 0 0
$$217$$ −10.0000 −0.678844
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 28.2843 1.90261
$$222$$ 0 0
$$223$$ −28.4605 −1.90586 −0.952928 0.303197i $$-0.901946\pi$$
−0.952928 + 0.303197i $$0.901946\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −13.4164 −0.890478 −0.445239 0.895412i $$-0.646881\pi$$
−0.445239 + 0.895412i $$0.646881\pi$$
$$228$$ 0 0
$$229$$ 13.4164 0.886581 0.443291 0.896378i $$-0.353811\pi$$
0.443291 + 0.896378i $$0.353811\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$234$$ 0 0
$$235$$ −16.9706 −1.10704
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 4.00000 0.258738 0.129369 0.991596i $$-0.458705\pi$$
0.129369 + 0.991596i $$0.458705\pi$$
$$240$$ 0 0
$$241$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −4.24264 −0.271052
$$246$$ 0 0
$$247$$ −12.6491 −0.804844
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −13.4164 −0.846836 −0.423418 0.905934i $$-0.639170\pi$$
−0.423418 + 0.905934i $$0.639170\pi$$
$$252$$ 0 0
$$253$$ 17.8885 1.12464
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 12.6491 0.789030 0.394515 0.918890i $$-0.370913\pi$$
0.394515 + 0.918890i $$0.370913\pi$$
$$258$$ 0 0
$$259$$ 14.1421 0.878750
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 16.0000 0.986602 0.493301 0.869859i $$-0.335790\pi$$
0.493301 + 0.869859i $$0.335790\pi$$
$$264$$ 0 0
$$265$$ 10.0000 0.614295
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 18.3848 1.12094 0.560470 0.828175i $$-0.310621\pi$$
0.560470 + 0.828175i $$0.310621\pi$$
$$270$$ 0 0
$$271$$ −3.16228 −0.192095 −0.0960473 0.995377i $$-0.530620\pi$$
−0.0960473 + 0.995377i $$0.530620\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −13.4164 −0.809040
$$276$$ 0 0
$$277$$ 13.4164 0.806114 0.403057 0.915175i $$-0.367948\pi$$
0.403057 + 0.915175i $$0.367948\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 12.6491 0.754583 0.377291 0.926095i $$-0.376856\pi$$
0.377291 + 0.926095i $$0.376856\pi$$
$$282$$ 0 0
$$283$$ 28.2843 1.68133 0.840663 0.541559i $$-0.182166\pi$$
0.840663 + 0.541559i $$0.182166\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 20.0000 1.18056
$$288$$ 0 0
$$289$$ 23.0000 1.35294
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −29.6985 −1.73500 −0.867502 0.497434i $$-0.834276\pi$$
−0.867502 + 0.497434i $$0.834276\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 17.8885 1.03452
$$300$$ 0 0
$$301$$ −26.8328 −1.54662
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 18.9737 1.08643
$$306$$ 0 0
$$307$$ −11.3137 −0.645707 −0.322854 0.946449i $$-0.604642\pi$$
−0.322854 + 0.946449i $$0.604642\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −8.00000 −0.453638 −0.226819 0.973937i $$-0.572833\pi$$
−0.226819 + 0.973937i $$0.572833\pi$$
$$312$$ 0 0
$$313$$ 16.0000 0.904373 0.452187 0.891923i $$-0.350644\pi$$
0.452187 + 0.891923i $$0.350644\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −9.89949 −0.556011 −0.278006 0.960579i $$-0.589673\pi$$
−0.278006 + 0.960579i $$0.589673\pi$$
$$318$$ 0 0
$$319$$ 18.9737 1.06232
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −17.8885 −0.995345
$$324$$ 0 0
$$325$$ −13.4164 −0.744208
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 37.9473 2.09210
$$330$$ 0 0
$$331$$ −33.9411 −1.86557 −0.932786 0.360429i $$-0.882630\pi$$
−0.932786 + 0.360429i $$0.882630\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −12.0000 −0.653682 −0.326841 0.945079i $$-0.605984\pi$$
−0.326841 + 0.945079i $$0.605984\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −14.1421 −0.765840
$$342$$ 0 0
$$343$$ −12.6491 −0.682988
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −4.47214 −0.240077 −0.120038 0.992769i $$-0.538302\pi$$
−0.120038 + 0.992769i $$0.538302\pi$$
$$348$$ 0 0
$$349$$ 4.47214 0.239388 0.119694 0.992811i $$-0.461809\pi$$
0.119694 + 0.992811i $$0.461809\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −25.2982 −1.34649 −0.673244 0.739420i $$-0.735100\pi$$
−0.673244 + 0.739420i $$0.735100\pi$$
$$354$$ 0 0
$$355$$ −11.3137 −0.600469
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 24.0000 1.26667 0.633336 0.773877i $$-0.281685\pi$$
0.633336 + 0.773877i $$0.281685\pi$$
$$360$$ 0 0
$$361$$ −11.0000 −0.578947
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 5.65685 0.296093
$$366$$ 0 0
$$367$$ 22.1359 1.15549 0.577743 0.816218i $$-0.303933\pi$$
0.577743 + 0.816218i $$0.303933\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −22.3607 −1.16091
$$372$$ 0 0
$$373$$ −22.3607 −1.15779 −0.578896 0.815401i $$-0.696516\pi$$
−0.578896 + 0.815401i $$0.696516\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 18.9737 0.977194
$$378$$ 0 0
$$379$$ 25.4558 1.30758 0.653789 0.756677i $$-0.273178\pi$$
0.653789 + 0.756677i $$0.273178\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 4.00000 0.204390 0.102195 0.994764i $$-0.467413\pi$$
0.102195 + 0.994764i $$0.467413\pi$$
$$384$$ 0 0
$$385$$ −20.0000 −1.01929
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 24.0416 1.21896 0.609480 0.792802i $$-0.291378\pi$$
0.609480 + 0.792802i $$0.291378\pi$$
$$390$$ 0 0
$$391$$ 25.2982 1.27939
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −4.47214 −0.225018
$$396$$ 0 0
$$397$$ −4.47214 −0.224450 −0.112225 0.993683i $$-0.535798\pi$$
−0.112225 + 0.993683i $$0.535798\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 18.9737 0.947500 0.473750 0.880659i $$-0.342900\pi$$
0.473750 + 0.880659i $$0.342900\pi$$
$$402$$ 0 0
$$403$$ −14.1421 −0.704470
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 20.0000 0.991363
$$408$$ 0 0
$$409$$ −26.0000 −1.28562 −0.642809 0.766027i $$-0.722231\pi$$
−0.642809 + 0.766027i $$0.722231\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 6.32456 0.310460
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 4.47214 0.218478 0.109239 0.994016i $$-0.465159\pi$$
0.109239 + 0.994016i $$0.465159\pi$$
$$420$$ 0 0
$$421$$ −40.2492 −1.96163 −0.980814 0.194948i $$-0.937546\pi$$
−0.980814 + 0.194948i $$0.937546\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −18.9737 −0.920358
$$426$$ 0 0
$$427$$ −42.4264 −2.05316
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −12.0000 −0.578020 −0.289010 0.957326i $$-0.593326\pi$$
−0.289010 + 0.957326i $$0.593326\pi$$
$$432$$ 0 0
$$433$$ 34.0000 1.63394 0.816968 0.576683i $$-0.195653\pi$$
0.816968 + 0.576683i $$0.195653\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −11.3137 −0.541208
$$438$$ 0 0
$$439$$ −22.1359 −1.05649 −0.528245 0.849092i $$-0.677150\pi$$
−0.528245 + 0.849092i $$0.677150\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 31.3050 1.48734 0.743672 0.668545i $$-0.233083\pi$$
0.743672 + 0.668545i $$0.233083\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −6.32456 −0.298474 −0.149237 0.988801i $$-0.547682\pi$$
−0.149237 + 0.988801i $$0.547682\pi$$
$$450$$ 0 0
$$451$$ 28.2843 1.33185
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −20.0000 −0.937614
$$456$$ 0 0
$$457$$ −22.0000 −1.02912 −0.514558 0.857455i $$-0.672044\pi$$
−0.514558 + 0.857455i $$0.672044\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 1.41421 0.0658665 0.0329332 0.999458i $$-0.489515\pi$$
0.0329332 + 0.999458i $$0.489515\pi$$
$$462$$ 0 0
$$463$$ 15.8114 0.734818 0.367409 0.930060i $$-0.380245\pi$$
0.367409 + 0.930060i $$0.380245\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 22.3607 1.03473 0.517364 0.855765i $$-0.326913\pi$$
0.517364 + 0.855765i $$0.326913\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −37.9473 −1.74482
$$474$$ 0 0
$$475$$ 8.48528 0.389331
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 24.0000 1.09659 0.548294 0.836286i $$-0.315277\pi$$
0.548294 + 0.836286i $$0.315277\pi$$
$$480$$ 0 0
$$481$$ 20.0000 0.911922
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −2.82843 −0.128432
$$486$$ 0 0
$$487$$ −9.48683 −0.429889 −0.214945 0.976626i $$-0.568957\pi$$
−0.214945 + 0.976626i $$0.568957\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −17.8885 −0.807299 −0.403649 0.914914i $$-0.632258\pi$$
−0.403649 + 0.914914i $$0.632258\pi$$
$$492$$ 0 0
$$493$$ 26.8328 1.20849
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 25.2982 1.13478
$$498$$ 0 0
$$499$$ −16.9706 −0.759707 −0.379853 0.925047i $$-0.624026\pi$$
−0.379853 + 0.925047i $$0.624026\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −16.0000 −0.713405 −0.356702 0.934218i $$-0.616099\pi$$
−0.356702 + 0.934218i $$0.616099\pi$$
$$504$$ 0 0
$$505$$ 22.0000 0.978987
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 9.89949 0.438787 0.219394 0.975636i $$-0.429592\pi$$
0.219394 + 0.975636i $$0.429592\pi$$
$$510$$ 0 0
$$511$$ −12.6491 −0.559564
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 13.4164 0.591198
$$516$$ 0 0
$$517$$ 53.6656 2.36021
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 6.32456 0.277084 0.138542 0.990357i $$-0.455758\pi$$
0.138542 + 0.990357i $$0.455758\pi$$
$$522$$ 0 0
$$523$$ −14.1421 −0.618392 −0.309196 0.950998i $$-0.600060\pi$$
−0.309196 + 0.950998i $$0.600060\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −20.0000 −0.871214
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 28.2843 1.22513
$$534$$ 0 0
$$535$$ 25.2982 1.09374
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 13.4164 0.577886
$$540$$ 0 0
$$541$$ −40.2492 −1.73045 −0.865225 0.501384i $$-0.832824\pi$$
−0.865225 + 0.501384i $$0.832824\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −6.32456 −0.270914
$$546$$ 0 0
$$547$$ −2.82843 −0.120935 −0.0604674 0.998170i $$-0.519259\pi$$
−0.0604674 + 0.998170i $$0.519259\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −12.0000 −0.511217
$$552$$ 0 0
$$553$$ 10.0000 0.425243
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −24.0416 −1.01868 −0.509338 0.860566i $$-0.670110\pi$$
−0.509338 + 0.860566i $$0.670110\pi$$
$$558$$ 0 0
$$559$$ −37.9473 −1.60500
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 4.47214 0.188478 0.0942390 0.995550i $$-0.469958\pi$$
0.0942390 + 0.995550i $$0.469958\pi$$
$$564$$ 0 0
$$565$$ 17.8885 0.752577
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −18.9737 −0.795417 −0.397709 0.917512i $$-0.630195\pi$$
−0.397709 + 0.917512i $$0.630195\pi$$
$$570$$ 0 0
$$571$$ 22.6274 0.946928 0.473464 0.880813i $$-0.343003\pi$$
0.473464 + 0.880813i $$0.343003\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −12.0000 −0.500435
$$576$$ 0 0
$$577$$ −12.0000 −0.499567 −0.249783 0.968302i $$-0.580359\pi$$
−0.249783 + 0.968302i $$0.580359\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −14.1421 −0.586715
$$582$$ 0 0
$$583$$ −31.6228 −1.30968
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −35.7771 −1.47668 −0.738339 0.674430i $$-0.764390\pi$$
−0.738339 + 0.674430i $$0.764390\pi$$
$$588$$ 0 0
$$589$$ 8.94427 0.368542
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −37.9473 −1.55831 −0.779155 0.626831i $$-0.784352\pi$$
−0.779155 + 0.626831i $$0.784352\pi$$
$$594$$ 0 0
$$595$$ −28.2843 −1.15954
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −36.0000 −1.47092 −0.735460 0.677568i $$-0.763034\pi$$
−0.735460 + 0.677568i $$0.763034\pi$$
$$600$$ 0 0
$$601$$ −28.0000 −1.14214 −0.571072 0.820900i $$-0.693472\pi$$
−0.571072 + 0.820900i $$0.693472\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −12.7279 −0.517464
$$606$$ 0 0
$$607$$ −9.48683 −0.385059 −0.192529 0.981291i $$-0.561669\pi$$
−0.192529 + 0.981291i $$0.561669\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 53.6656 2.17108
$$612$$ 0 0
$$613$$ 13.4164 0.541884 0.270942 0.962596i $$-0.412665\pi$$
0.270942 + 0.962596i $$0.412665\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −25.2982 −1.01847 −0.509234 0.860628i $$-0.670071\pi$$
−0.509234 + 0.860628i $$0.670071\pi$$
$$618$$ 0 0
$$619$$ −11.3137 −0.454736 −0.227368 0.973809i $$-0.573012\pi$$
−0.227368 + 0.973809i $$0.573012\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −1.00000 −0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 28.2843 1.12777
$$630$$ 0 0
$$631$$ 41.1096 1.63655 0.818274 0.574829i $$-0.194931\pi$$
0.818274 + 0.574829i $$0.194931\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −22.3607 −0.887357
$$636$$ 0 0
$$637$$ 13.4164 0.531577
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 6.32456 0.249805 0.124902 0.992169i $$-0.460138\pi$$
0.124902 + 0.992169i $$0.460138\pi$$
$$642$$ 0 0
$$643$$ −8.48528 −0.334627 −0.167313 0.985904i $$-0.553509\pi$$
−0.167313 + 0.985904i $$0.553509\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 12.0000 0.471769 0.235884 0.971781i $$-0.424201\pi$$
0.235884 + 0.971781i $$0.424201\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 1.41421 0.0553425 0.0276712 0.999617i $$-0.491191\pi$$
0.0276712 + 0.999617i $$0.491191\pi$$
$$654$$ 0 0
$$655$$ −25.2982 −0.988483
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 17.8885 0.696839 0.348419 0.937339i $$-0.386719\pi$$
0.348419 + 0.937339i $$0.386719\pi$$
$$660$$ 0 0
$$661$$ −4.47214 −0.173946 −0.0869730 0.996211i $$-0.527719\pi$$
−0.0869730 + 0.996211i $$0.527719\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 12.6491 0.490511
$$666$$ 0 0
$$667$$ 16.9706 0.657103
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −60.0000 −2.31627
$$672$$ 0 0
$$673$$ 34.0000 1.31060 0.655302 0.755367i $$-0.272541\pi$$
0.655302 + 0.755367i $$0.272541\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 38.1838 1.46752 0.733761 0.679408i $$-0.237763\pi$$
0.733761 + 0.679408i $$0.237763\pi$$
$$678$$ 0 0
$$679$$ 6.32456 0.242714
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 40.2492 1.54009 0.770047 0.637987i $$-0.220233\pi$$
0.770047 + 0.637987i $$0.220233\pi$$
$$684$$ 0 0
$$685$$ 26.8328 1.02523
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −31.6228 −1.20473
$$690$$ 0 0
$$691$$ 8.48528 0.322795 0.161398 0.986889i $$-0.448400\pi$$
0.161398 + 0.986889i $$0.448400\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 24.0000 0.910372
$$696$$ 0 0
$$697$$ 40.0000 1.51511
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −26.8701 −1.01487 −0.507434 0.861691i $$-0.669406\pi$$
−0.507434 + 0.861691i $$0.669406\pi$$
$$702$$ 0 0
$$703$$ −12.6491 −0.477070
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −49.1935 −1.85011
$$708$$ 0 0
$$709$$ 13.4164 0.503864 0.251932 0.967745i $$-0.418934\pi$$
0.251932 + 0.967745i $$0.418934\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −12.6491 −0.473713
$$714$$ 0 0
$$715$$ −28.2843 −1.05777
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$720$$ 0 0
$$721$$ −30.0000 −1.11726
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −12.7279 −0.472703
$$726$$ 0 0
$$727$$ −15.8114 −0.586412 −0.293206 0.956049i $$-0.594722\pi$$
−0.293206 + 0.956049i $$0.594722\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −53.6656 −1.98490
$$732$$ 0 0
$$733$$ −22.3607 −0.825911 −0.412955 0.910751i $$-0.635503\pi$$
−0.412955 + 0.910751i $$0.635503\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 45.2548 1.66473 0.832363 0.554231i $$-0.186988\pi$$
0.832363 + 0.554231i $$0.186988\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −44.0000 −1.61420 −0.807102 0.590412i $$-0.798965\pi$$
−0.807102 + 0.590412i $$0.798965\pi$$
$$744$$ 0 0
$$745$$ 26.0000 0.952566
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −56.5685 −2.06697
$$750$$ 0 0
$$751$$ 15.8114 0.576966 0.288483 0.957485i $$-0.406849\pi$$
0.288483 + 0.957485i $$0.406849\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −31.3050 −1.13930
$$756$$ 0 0
$$757$$ −40.2492 −1.46288 −0.731441 0.681904i $$-0.761152\pi$$
−0.731441 + 0.681904i $$0.761152\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 18.9737 0.687795 0.343897 0.939007i $$-0.388253\pi$$
0.343897 + 0.939007i $$0.388253\pi$$
$$762$$ 0 0
$$763$$ 14.1421 0.511980
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −40.0000 −1.44244 −0.721218 0.692708i $$-0.756418\pi$$
−0.721218 + 0.692708i $$0.756418\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 29.6985 1.06818 0.534090 0.845428i $$-0.320654\pi$$
0.534090 + 0.845428i $$0.320654\pi$$
$$774$$ 0 0
$$775$$ 9.48683 0.340777
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −17.8885 −0.640924
$$780$$ 0 0
$$781$$ 35.7771 1.28020
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −18.9737 −0.677199
$$786$$ 0 0
$$787$$ 42.4264 1.51234 0.756169 0.654376i $$-0.227069\pi$$
0.756169 + 0.654376i $$0.227069\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −40.0000 −1.42224
$$792$$ 0 0
$$793$$ −60.0000 −2.13066
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −18.3848 −0.651222 −0.325611 0.945504i $$-0.605570\pi$$
−0.325611 + 0.945504i $$0.605570\pi$$
$$798$$ 0 0
$$799$$ 75.8947 2.68496
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −17.8885 −0.631273
$$804$$ 0 0
$$805$$ −17.8885 −0.630488
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 18.9737 0.667079 0.333539 0.942736i $$-0.391757\pi$$
0.333539 + 0.942736i $$0.391757\pi$$
$$810$$ 0 0
$$811$$ 19.7990 0.695237 0.347618 0.937636i $$-0.386991\pi$$
0.347618 + 0.937636i $$0.386991\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −28.0000 −0.980797
$$816$$ 0 0
$$817$$ 24.0000 0.839654
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 12.7279 0.444208 0.222104 0.975023i $$-0.428708\pi$$
0.222104 + 0.975023i $$0.428708\pi$$
$$822$$ 0 0
$$823$$ 15.8114 0.551150 0.275575 0.961280i $$-0.411132\pi$$
0.275575 + 0.961280i $$0.411132\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$828$$ 0 0
$$829$$ −13.4164 −0.465971 −0.232986 0.972480i $$-0.574849\pi$$
−0.232986 + 0.972480i $$0.574849\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 18.9737 0.657399
$$834$$ 0 0
$$835$$ −16.9706 −0.587291
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −16.0000 −0.552381 −0.276191 0.961103i $$-0.589072\pi$$
−0.276191 + 0.961103i $$0.589072\pi$$
$$840$$ 0 0
$$841$$ −11.0000 −0.379310
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −9.89949 −0.340553
$$846$$ 0 0
$$847$$ 28.4605 0.977914
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 17.8885 0.613211
$$852$$ 0 0
$$853$$ 13.4164 0.459369 0.229685 0.973265i $$-0.426231\pi$$
0.229685 + 0.973265i $$0.426231\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 44.2719 1.51230 0.756149 0.654399i $$-0.227078\pi$$
0.756149 + 0.654399i $$0.227078\pi$$
$$858$$ 0 0
$$859$$ 25.4558 0.868542 0.434271 0.900782i $$-0.357006\pi$$
0.434271 + 0.900782i $$0.357006\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −36.0000 −1.22545 −0.612727 0.790295i $$-0.709928\pi$$
−0.612727 + 0.790295i $$0.709928\pi$$
$$864$$ 0 0
$$865$$ −10.0000 −0.340010
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 14.1421 0.479739
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 35.7771 1.20949
$$876$$ 0 0
$$877$$ −13.4164 −0.453040 −0.226520 0.974007i $$-0.572735\pi$$
−0.226520 + 0.974007i $$0.572735\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −12.6491 −0.426159 −0.213080 0.977035i $$-0.568349\pi$$
−0.213080 + 0.977035i $$0.568349\pi$$
$$882$$ 0 0
$$883$$ −19.7990 −0.666289 −0.333145 0.942876i $$-0.608110\pi$$
−0.333145 + 0.942876i $$0.608110\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −32.0000 −1.07445 −0.537227 0.843437i $$-0.680528\pi$$
−0.537227 + 0.843437i $$0.680528\pi$$
$$888$$ 0 0
$$889$$ 50.0000 1.67695
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −33.9411 −1.13580
$$894$$ 0 0
$$895$$ −25.2982 −0.845626
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −13.4164 −0.447462
$$900$$ 0 0
$$901$$ −44.7214 −1.48988
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −18.9737 −0.630706
$$906$$ 0 0
$$907$$ 14.1421 0.469582 0.234791 0.972046i $$-0.424559\pi$$
0.234791 + 0.972046i $$0.424559\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −12.0000 −0.397578 −0.198789 0.980042i $$-0.563701\pi$$
−0.198789 + 0.980042i $$0.563701\pi$$
$$912$$ 0 0
$$913$$ −20.0000 −0.661903
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 56.5685 1.86806
$$918$$ 0 0
$$919$$ −34.7851 −1.14745 −0.573727 0.819047i $$-0.694503\pi$$
−0.573727 + 0.819047i $$0.694503\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 35.7771 1.17762
$$924$$ 0 0
$$925$$ −13.4164 −0.441129
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 56.9210 1.86752 0.933759 0.357903i $$-0.116508\pi$$
0.933759 + 0.357903i $$0.116508\pi$$
$$930$$ 0 0
$$931$$ −8.48528 −0.278094
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ −40.0000 −1.30814
$$936$$ 0 0
$$937$$ 22.0000 0.718709 0.359354 0.933201i $$-0.382997\pi$$
0.359354 + 0.933201i $$0.382997\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 57.9828 1.89018 0.945092 0.326805i $$-0.105972\pi$$
0.945092 + 0.326805i $$0.105972\pi$$
$$942$$ 0 0
$$943$$ 25.2982 0.823823
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 17.8885 0.581300 0.290650 0.956830i $$-0.406129\pi$$
0.290650 + 0.956830i $$0.406129\pi$$
$$948$$ 0 0
$$949$$ −17.8885 −0.580687
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −18.9737 −0.614617 −0.307309 0.951610i $$-0.599428\pi$$
−0.307309 + 0.951610i $$0.599428\pi$$
$$954$$ 0 0
$$955$$ −28.2843 −0.915258
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −60.0000 −1.93750
$$960$$ 0 0
$$961$$ −21.0000 −0.677419
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 22.6274 0.728402
$$966$$ 0 0
$$967$$ −41.1096 −1.32200 −0.660998 0.750388i $$-0.729867\pi$$
−0.660998 + 0.750388i $$0.729867\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 22.3607 0.717588 0.358794 0.933417i $$-0.383188\pi$$
0.358794 + 0.933417i $$0.383188\pi$$
$$972$$ 0 0
$$973$$ −53.6656 −1.72044
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 56.9210 1.82106 0.910532 0.413439i $$-0.135672\pi$$
0.910532 + 0.413439i $$0.135672\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 24.0000 0.765481 0.382741 0.923856i $$-0.374980\pi$$
0.382741 + 0.923856i $$0.374980\pi$$
$$984$$ 0 0
$$985$$ 10.0000 0.318626
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −33.9411 −1.07927
$$990$$ 0 0
$$991$$ −34.7851 −1.10498 −0.552492 0.833518i $$-0.686323\pi$$
−0.552492 + 0.833518i $$0.686323\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −4.47214 −0.141776
$$996$$ 0 0
$$997$$ 22.3607 0.708170 0.354085 0.935213i $$-0.384792\pi$$
0.354085 + 0.935213i $$0.384792\pi$$
$$998$$ 0 0
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 4608.2.a.y.1.2 yes 4
3.2 odd 2 4608.2.a.x.1.4 yes 4
4.3 odd 2 4608.2.a.x.1.1 4
8.3 odd 2 4608.2.a.x.1.3 yes 4
8.5 even 2 inner 4608.2.a.y.1.4 yes 4
12.11 even 2 inner 4608.2.a.y.1.3 yes 4
16.3 odd 4 4608.2.d.l.2305.4 4
16.5 even 4 4608.2.d.i.2305.1 4
16.11 odd 4 4608.2.d.l.2305.2 4
16.13 even 4 4608.2.d.i.2305.3 4
24.5 odd 2 4608.2.a.x.1.2 yes 4
24.11 even 2 inner 4608.2.a.y.1.1 yes 4
48.5 odd 4 4608.2.d.l.2305.3 4
48.11 even 4 4608.2.d.i.2305.4 4
48.29 odd 4 4608.2.d.l.2305.1 4
48.35 even 4 4608.2.d.i.2305.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
4608.2.a.x.1.1 4 4.3 odd 2
4608.2.a.x.1.2 yes 4 24.5 odd 2
4608.2.a.x.1.3 yes 4 8.3 odd 2
4608.2.a.x.1.4 yes 4 3.2 odd 2
4608.2.a.y.1.1 yes 4 24.11 even 2 inner
4608.2.a.y.1.2 yes 4 1.1 even 1 trivial
4608.2.a.y.1.3 yes 4 12.11 even 2 inner
4608.2.a.y.1.4 yes 4 8.5 even 2 inner
4608.2.d.i.2305.1 4 16.5 even 4
4608.2.d.i.2305.2 4 48.35 even 4
4608.2.d.i.2305.3 4 16.13 even 4
4608.2.d.i.2305.4 4 48.11 even 4
4608.2.d.l.2305.1 4 48.29 odd 4
4608.2.d.l.2305.2 4 16.11 odd 4
4608.2.d.l.2305.3 4 48.5 odd 4
4608.2.d.l.2305.4 4 16.3 odd 4